**1. Introduction**

The compressible Navier–Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interests. They are widely used but not limited to the fields of fluid dynamics, aeronautical engineering, and aerodynamics engineering. The Navier–Stokes equations are certain partial differential equations that describe the motion of viscous fluid substances. The compressible Navier–Stokes equations mathematically express the continuity, the theorem of momentum, and the theorem of energy of Newtonian fluids. These equations arise from applying Isaac Newton's second law to fluid motion, with the assumption that the stress in the fluid is the sum of a diffusing viscous term and a pressure term. The diffusing viscous term is proportional to the gradient of velocity. The difference between Euler equations and Navier–Stokes equations is that Euler equations are only valid for inviscid flow, while Navier–Stokes equations take viscosity into account. Euler equations are hyperbolic, while Navier–Stokes equations are parabolic.

Although the discontinuous Galerkin (DG) method is a natural choice for hyperbolic problems, it becomes far less certain when it comes to elliptic or parabolic problems. An alternative approach for viscous discretization, which reformulates the viscous terms as a first-order hyperbolic system (FOHS), was given in Refs. [1, 2]. Another FOHS formulation [3, 4] was developed by including the gradient quantities as additional variables. Over several years of development, the FOHS method has been shown to offer several distinguished advantages. First, the well-developed numerical schemes for hyperbolic systems can be directly applied to viscous problems. Second, it enables high-quality gradient prediction even on fully irregular grids. This feature shows its significance when it comes to viscous flow simulations on complex grids, where the qualities of the meshes are likely to be highly irregular. Third, due to the fact that the second-order derivatives in the governing partial differential equations (PDE) are replaced and eliminated, the developed scheme shows speedup and robustness for the iterative solvers. Finally, the FOHS method can improve viscous discretization as well as inviscid discretization. Due to these favorable characteristics, the hyperbolic methods have been implemented in various applications, including diffusion [5], anisotropic diffusion [6], advection–diffusion [7], Navier-Stokes (NS) equations [8], and three-dimensional compressible NS equations with proper handling of high Reynolds number boundary layer flows [8].

To make it easier for the readers to understand, the classical computation methods for fluid dynamics are first discussed, and the reconstructed discontinuous Galerkin (rDG) method is introduced. FOHS for linear advection–diffusion equation is then given. Eventually, the hyperbolic Navier–Stokes with rDG method is given.
